NFW Dark Matter Halos and Virial Radius

[zephyr5050]

I've been working with NFW Dark Matter Halos recently. This is a particular density model for the halo developed by Navarro, Frenk, & White (NFW). The density structure has the form
$$\rho (r) = \frac{\delta_c \rho_c}{(r/r_s)(1+r/r_s)^2}$$
where
$$\delta_c = \frac{200}{3} \frac{c^3}{ln(1+c)-c/(1+c)}$$
$$r_s = r_{200}/c$$
and $\rho_c$ is the critical density of the universe (as a function of redshift). The parameter $r_{200}$ is the virial radius which is defined as the radius at which the mass density of the halo is $200\rho_c$.

Now we can't really talk about the mass of this halo because the integral from 0 to $\infty$ diverges. Instead, we use the fiducial radius $r_{200}$ and define the quantity $M_{200}$ to be the mass inside the radius $r_{200}$. It can be shown that
$$M_{200} = \frac{800\pi}{3}\rho_c r_{200}^3$$
While all this makes sense to me, there's one thing that I don't understand here. Where does this 200 come from? Why say $r_{200} \equiv 200 \rho_c$? Is there any logic to this, is it historical, arbitrary? What's going on here?

 

[phyzguy]

I think it's basically historical and arbitrary. You have to draw a line somewhere, and the accepted way to do it is when the average density of the cluster falls to 200X the critical density. It's not universal, however. You will find papers referring to R500, R100, R150, etc., all defined in the same way, but with different multipliers.

 

[zephyr5050]

Do you happen to know the paper which proposed this commonly accepted line? If any?

 

[phyzguy]

I'm not sure, but I suggest starting with the original NFW paper:

http://arxiv.org/pdf/astro-ph/9508025.pdf

 

[sardar]

I can provide some insight into the reasoning behind using the value of 200 for the virial radius in the NFW dark matter halo model.

The value of 200 comes from the fact that it is a common threshold used to define the virial radius in cosmological simulations and observations. This threshold is based on the assumption that the dark matter halos are in approximate virial equilibrium, meaning that the gravitational forces within the halo are balanced by the kinetic energy of the particles within it.

In other words, the virial radius is the radius at which the average density of the halo is 200 times the critical density of the universe. This value is chosen because it corresponds to the point at which the gravitational potential energy of the halo is equal to its kinetic energy, indicating virial equilibrium.

The use of this threshold is not arbitrary or historical, but rather a practical choice for defining the virial radius in a consistent manner across different simulations and observations. It also allows for a more manageable and meaningful measure of the mass of the halo, as the integral from 0 to infinity diverges when trying to calculate the total mass of the halo.

Overall, the use of 200 for the virial radius in the NFW dark matter halo model is a well-established and widely accepted convention in the field of cosmology.